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Práctica de ejercicios de cálculo de tema de derivada de funciones
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  • Course
  • Cálculo Diferencial
  • Institution
  • Universidad Nacional De San Martín

En este documento se presenta ejercicios desarrollados de problemas matemáticos del tema de derivada de funciones, la cual es un campo muy importante del área de las matemáticas.

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  • June 1, 2021
  • 8
  • 2020/2021
  • Other
  • Unknown

Subjects

  • matemática
  • cálculo
  • avanzado
  • universidad
  • derivada
  • funciones
  • derivada de funciones
  • números
  • ciencia exacta
  • exacto
  • práctica
  • ejercicios

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  • Universidad Nacional de San Martín
  • Cálculo Diferencial
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UNSM
Universidad Nacional de San Martín




FACULTAD DE CIENCIAS AGRARIAS
ESCUELA PROFESIONAL DE AGRONOMIA


INFORME N° 03

Ciclo:
II

Asignatura:
Análisis Matemático

Unidad:
III

Docente:
Dr. Ing. Fernando Ruiz Saavedra

Tema:
Derivada

Estudiantes:
Rubén Vásquez Bazán
Jesús Xavier Quintos Livaque
Herlis Esduvil Córdova Paucar
Miguel Ángel Campos Valera
Lucas Augusto Panduro Ríos




Tarapoto - Perú
2021

, S O LU C I O N AR I O
Derivadas 11. 𝒉(𝒙) = (𝒙𝟐 − 𝟏)𝟐
************************************************ Desarrollo:
Práctica I ℎ′(𝑥) = 2(𝑥 2 − 1) 2−1
Encontrar la derivada de las siguientes funciones: ℎ′(𝑥) = 2(𝑥 2 − 1)(2𝑥)
1. 𝑭(𝒙) = 𝟕𝒙𝟒 − 𝟐𝒙𝟑 + 𝟖𝒙 + 𝟓 ℎ′(𝑥) = 4𝑥(𝑥 2 − 1)
Desarrollo: ∴ 𝒉′(𝒙) = 𝟒𝒙𝟑 − 𝟒𝒙
𝐹′(𝑥) = 7.4𝑥 4−1 − 2.3𝑥 3−1 + 1.8𝑥1−1 + 0 12. 𝒉′(𝒔 ) = (𝒔 𝟐 − 𝟐) 𝟐
∴ 𝑭′ (𝒙) = 𝟐𝟖𝒙𝟑 − 𝟔𝒙𝟐 + 𝟖 Desarrollo:
2. 𝒚 = 𝟑𝒙−𝟒 + 𝟑𝒙𝟒 ℎ′(𝑠) = 2(𝑠 2 − 2) 2−1
Desarrollo: ℎ′(𝑠) = 2(𝑠 2 − 2) (2𝑠)
𝑦 ′ = −12𝑥 −5 + 12𝑥 3 ℎ′(𝑠) = 4𝑠 (𝑠 2 − 2)
12
𝑦 ′ = − 5 + 12𝑥 3 ∴ 𝒉′ (𝒔 ) = 𝟒𝒔 𝟑 − 𝟖𝒔
𝑥
−𝟏𝟐+𝟏𝟐𝒙𝟖 13. 𝑭(𝒙) = (𝒙𝟐 − 𝒙)(𝒙𝟐 + 𝟏)(𝒙𝟐 + 𝒙 + 𝟏)
′
∴𝒚 = Desarrollo
𝒙𝟓
3. 𝒉(𝒙) = (𝟐𝒙𝟑 − 𝟒𝒙𝟐 )(𝟑𝒙𝟓 + 𝒙𝟐 ) 𝐹(𝑥) = 𝑥 6 − 𝑥 2 + 𝑥 3 + 𝑥 2
Desarrollo: ∴ 𝑭′(𝒙) = 𝟔𝒙𝟓 − 𝟐𝒙 + 𝟑𝒙𝟐 + 𝟐𝒙
ℎ′(𝑥) = (2𝑥3 − 4𝑥2)(15𝑥4 + 2𝑥) + (6𝑥2 − 8𝑥)(3𝑥5 + 𝑥2 ) 14. 𝑭(𝒙) = (𝟑𝒙𝟑 + 𝟒𝒙)(𝒙 − 𝟓)(𝒙 + 𝟏)
ℎ′(𝑥) = 30𝑥7 + 4𝑥4 − 60𝑥6 − 8𝑥3 + 18𝑥7 − 24𝑥6 + 6𝑥4 − 8𝑥3 Desarrollo:
∴ 𝒉′(𝒙) = 𝟒𝟖𝒙𝟕 − 𝟖𝟒𝒙𝟔 + 𝟏𝟎𝒙𝟒 − 𝟏𝟔𝒙𝟑 𝐹(𝑥) = 3𝑥 5 − 12𝑥 4 − 11𝑥 3 − 16𝑥 2 − 20𝑥
4. 𝑭(𝒙) = (𝟑𝒙 − 𝟐𝒙𝟐 )(𝟓 + 𝟒𝒙) ∴ 𝑭′(𝒙) = 𝟏𝟓𝒙𝟒 − 𝟒𝟖𝒙𝟑 − 𝟑𝟑𝒙𝟐 − 𝟑𝟐𝒙 − 𝟐𝟎
Desarrollo:
15. 𝒚 = (𝟐𝒙𝟐 )(√𝟐− 𝒙)
𝐹′(𝑥) = ( 3𝑥 − 2𝑥 2 )( 4) + ( 3 − 4𝑥) ( 5 + 4𝑥)
𝐹′(𝑥) = 12𝑥 − 8𝑥 2 + 15 − 20𝑥 + 12𝑥 − 16𝑥 2
Desarrollo:
∴ 𝑭′( 𝒙) = −𝟐𝟒𝒙𝟐 + 𝟒𝒙 + 𝟏𝟓 𝑦 = √4𝑥 4. √ 2 − 𝑥
5. 𝒚 = (𝟏 + 𝒙−𝟏 )(𝒙 − 𝟏) 𝑦 = √8𝑥 4 − 4𝑥 5 → 𝑢 = 8𝑥 4 − 4𝑥 5
Desarrollo: 𝑑 𝑑
𝑦 ′ = √ 𝑢. (8𝑥 4 − 4𝑥 5 )
𝑦 = 𝑥 − 1 + 1 − 𝑥 −1 𝑑𝑢
1
𝑑𝑥

𝑦 = 𝑥 − 𝑥 −1 𝑦′ = (32𝑥 3 − 20𝑥 4 )
2√𝑢
𝑦 ′ = 1 + 𝑥 −2 𝑦′ =
1
(32𝑥 3 − 20𝑥 4 )
𝟏 2√ 8𝑥4−4𝑥5
∴ 𝒚′ = 𝟏 + 𝟐 32𝑥3 −20𝑥4
𝒙 𝑦′ =
6. 𝑭(𝒙) = (𝒙𝟐 − 𝟐𝒙 + 𝟏)(𝒙 − 𝟏) 4𝑥2 √2−𝑥
′ 𝟖𝒙−𝟓𝒙𝟐
Desarrollo: ∴𝒚 =
√𝟐−𝒙
𝐹(𝑥) = 𝑥 5 − 2𝑥 4 + 𝑥 3 − 𝑥 2 + 2𝑥 − 1
16. 𝑭(𝒙) = (𝒙)(√𝟑 − 𝟐𝒙𝟐 )
∴ 𝑭′ (𝒙) = 𝟓𝒙𝟒 − 𝟖𝒙𝟑 + 𝟑𝒙𝟐 − 𝟐𝒙 + 𝟐
Desarrollo:
7. 𝑭(𝒙) = (𝒙𝟑 − 𝟑𝒙)(𝟐𝒙𝟐 + 𝟑𝒙 + 𝟓)
Desarrollo: 𝐹(𝑥) = √𝑥2 √3 − 2𝑥 2
𝐹( 𝑥) = 2𝑥 5 + 3𝑥 4 + 5𝑥 3 − 6𝑥 3 − 9𝑥 2 − 15𝑥 𝐹(𝑥) = √3𝑥2 − 2𝑥 4 → 𝑢 = 3𝑥 2 − 2𝑥 4
𝑑 𝑑
𝐹( 𝑥) = 2𝑥 5 + 3𝑥 4 − 𝑥 3 − 9𝑥 2 − 15𝑥 𝐹′(𝑥) = √ 𝑢. (3𝑥 2 − 2𝑥 4 )
𝑑𝑢 𝑑𝑥
∴ 𝑭′(𝒙) = 𝟏𝟎𝒙𝟒 + 𝟏𝟐𝒙𝟑 − 𝟑𝒙𝟐 − 𝟏𝟖𝒙 − 𝟏𝟓 1
8. 𝑭(𝒙) = (𝒙 − 𝟏)(𝒙𝟐 − 𝟑𝒙 + 𝟐) 𝐹′(𝑥) = . (6𝑥 − 8𝑥 3 )
2√𝑢
Desarrollo: 6𝑥−8𝑥3
𝐹′(𝑥) =
𝐹(𝑥) = 𝑥 3 − 4𝑥 2 + 5𝑥 − 2 2√ 3𝑥2−2𝑥4
𝟑𝒙−𝟖𝒙𝟑
∴ 𝑭′(𝒙) = 𝟑𝒙𝟐 − 𝟖𝒙 + 𝟓 ∴ 𝑭′ (𝒙) =
𝟓 𝟏 |𝒙|√ 𝟑−𝟐𝒙𝟐
9. 𝑭(𝒙) = (𝒙 − 𝟑𝒙)( 𝟐)
𝒙 17. 𝒉(𝒙) = (𝟐𝒙 − 𝟒𝒙 (𝟑𝒙𝟓 + 𝒙𝟐 )
𝟑 𝟐)
Desarrollo: Desarrollo:
1
𝐹(𝑥) = 𝑥(𝑥 4 − 3). ℎ(𝑥) = 2𝑥 3(3𝑥 5 + 𝑥 2 ) − 4𝑥 2 (3𝑥 5 + 𝑥 2 )
𝑥2
𝑥4−3 ℎ(𝑥) = 6𝑥 8 + 2𝑥 5 − 12𝑥 7 − 4𝑥 4
𝐹(𝑥) =
𝑥 ∴ 𝒉′ (𝒙) = 𝟒𝟖𝒙𝟕 − 𝟖𝟒𝒙𝟔 + 𝟏𝟎𝒙𝟒 − 𝟏𝟔𝒙𝟑
𝑥4 3
𝐹(𝑥) = − 18. 𝒚 = (𝒙𝟐 + 𝟏𝟕)(𝒙𝟑 − 𝟑𝒙 + 𝟏)
𝑥 𝑥
𝑑𝐹 2 1 Desarrollo:
= 3𝑥 − (−3. 2 )
𝑑𝑥 𝑥 𝑦 = 𝑥 2 (𝑥 3 − 3𝑥 + 1) + 17(𝑥 3 − 3𝑥 + 1)
𝒅𝑭 𝟑𝒙𝟒+𝟑
∴ = 𝟐 𝑦 = 𝑥 5 − 3𝑥 3 + 𝑥 2 + 17𝑥 3 − 51𝑥 + 17
𝒅𝒙 𝒙
10. 𝑭(𝒙) = 𝟑√ 𝒙(√𝒙 + 𝟑) 𝑦 = 𝑥 5 + 14𝑥 3 + 𝑥 2 − 51𝑥 + 17
∴ 𝒚′ = 𝟓𝒙𝟒 + 𝟒𝟐𝒙𝟐 + 𝟐𝒙 − 𝟓𝟏
Desarrollo:
1 1 19. 𝒚 = (𝒙𝟒 + 𝟐𝒙)(𝒙𝟑 + 𝟐𝒙𝟐 + 𝟏)
𝐹(𝑥) = 𝑥 3 (𝑥 2 + 3) Desarrollo:
5 1
𝑦 = 𝑥 4 (𝑥 3 + 2𝑥 2 + 1) + 2𝑥(𝑥 3 + 2𝑥 2 + 1)
𝐹(𝑥) = 𝑥 6 + 3𝑥 3
5 1 1 2 𝑦 = 𝑥 7 + 2𝑥 6 + 𝑥 4 + 2𝑥 4 + 4𝑥 3 + 2𝑥
𝐹′(𝑥) = 𝑥 − 6 + (3) 𝑥 − 3 𝑦 = 𝑥 7 + 2𝑥 6 + 3𝑥 4 + 4𝑥 3 + 2𝑥
6 3
∴ 𝑭′ (𝒙) =
𝟓
+𝟑
𝟏 𝑦 = 7𝑥 7−1 + 6.2𝑥 6−1 + 4.3𝑥 4−1 + 3.4𝑥3−1 + (2𝑥)′
𝟔 𝟔√𝒙 √ 𝒙𝟐 ∴ 𝒚′ = 𝟕𝒙𝟔 + 𝟏𝟐𝒙𝟓 + 𝟏𝟐𝒙𝟑 + 𝟏𝟐𝒙𝟐 + 𝟐

Matemática 2021-1

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